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joined the club for the sole purpose of one question

need some checking on this formula

Hey everyone, this is off-topic for math, but curious if anyone's tried those online <a href="https://realiqtestonline.com/">IQ brain tests</a>? Are they accurate at all?

A rough estimate in some cases.

If your "arrow squared" (↑²) notation is supposed to mean the double arrow notation,  ↑↑, 
then I wonder,

a) why don't you use the usual ↑↑ notation? (obviously to be able to say "no, I didn't mean that"...)
(Note, it's a very common and strong (and commonly strongly agreed upon) crackpot flag to use nonstandard notation. *Absolutely* to be avoided if you want that people look at what you write.)

b) what it can mean for non-integer x, because it is usually defined as "iterate exponentiation x-1 times, e.g., 2 ↑↑ 4 = 2^(2^(2^2)). I don't see how we can (unambiguously) define that for a non-integer x. (Edit: I have thought a while... I found an idea how to define a↑↑(b+x) for 0 <= x <= 1, namely, you take  a↑↑(b+1) but replace the last a on top of the tower by a^x -- then for x=0 this gives 1, as if nothing was there in the exponent, and therefore a↑↑b, as required ; for x=1 it is there, as expected for a↑↑(b+1). I wonder whether this is the non-integer generalization you found somewhere - and if so, where?)

Obviously, we have first to define the generalisation to non-integer x's, before we can apply usual differentiation d/dx. Otherwise one could also look, e.g., at finite differences.

Actually : what is "n", appearing on the RHS but not on the LHS? 🤔

I agree. For more on this topic, please use our "IQ-Forum" (click here), let's limit the discussion here to topics around the double-arrow notation. - Thanks!